The effects of core-mantle gravitational coupling on the rotational dynamics of Mercury

As Mercury orbits the Sun, solar induced gravitational torques give rise to a planetary libration, which has been detected by Earth based radar speckle patterns. The amplitude of this libration suggests that only the mantle participates in the libration motion, thereby indicating a decoupling with the core. This is seen as proof that the outermost part of the core is fluid. While the planet undergoes its 88 day period libration, the axes of minimum moment of inertia of the mantle and solid core, if present, become misaligned, leading to a gravitational torque between the two. This initiates a free-mode of axial oscillation between the inner core and mantle. For a large gravitational torque, the free-mode period approaches the period of the libration forcing, and should participate in the planet's libration response. The goal of this work is to determine whether Mercury's observed librations can be used to place constraints on the size of its inner core. We model Mercury with three concentric layers, including a solid mantle and fluid and solid cores. By numerically solving the governing set of coupled differential equations, perturbations in Mercury's rotation rate are simulated for a range of interior structures. The model response is compared to spin rate observations. For models where the free-mode interferes with the libration signature, a marginally better fit between model response and observations is obtained, compared to models which exhibit the libration signature alone. We show that the best fit to observations occurs for a moment of inertia ratio (B_{m-A_m)/(C_m)} = (2.19 ± 0.04) × 10^-4 and inner core radius r_s>500 km.